You should be interested in the discussion in
EvansEvans, Gareth A.; Wensley, Christopher D., G.~A. and WensleyComplete involutive rewriting systems, C.~D. {Complete involutive rewriting systems}J. {J Symb. Symbolic Comput.} \textbf{42}~(11 42, No. 11-12), 1034-1051 (2007) 1034--1051. ZBL1128.68044.
on the notion of noncommutative involutive systems, which in the commutative case are a well known modification of the GrobnerGröbner basis theory. Evans' thesis is available at math.RA/0602140math.RA/0602140.
I can't help on the proposed applications.
Ronnie Brown